Route to chaos in a whispering gallery mode coupled opto-mechanical system

Abstract

In opto-mechanical systems, the nonlinearity caused by radiation pressure can lead to chaos and other abundant dynamical phenomena. Chaos is an important branch of nonlinear dynamics, and the routes from order to chaos in various systems are the research focus for academics. In this paper, we investigate the chaotic dynamics in a system consisting of two evanescently coupled identical cavity opto-mechanical subsystems, in which the optical fields are in the whispering gallery modes. In order to thoroughly investigate the transition from order to chaos in our system, we utilize the bifurcation diagram, the Lyapunov exponents and the phase space trajectories to characterize the system properties. It is found that the coupling strength between the two opto-mechanical subsystems is an important factor in determining the system dynamic behaviors. There are two routes to chaos in our system, that is, the period-doubling transition and the quasiperiodic transition, and they correspond to the cases of a strong and weak coupling between the two opto-mechanical subsystems, respectively. The results also show that the synchronization between the oscillations in the two opto-mechanical subsystems can take place in the case of strong coupling. In this situation, the dynamics of two opto-mechanical subsystems are exactly the same and the manipulation of the coupling strength is equivalent to tune the frequency detuning between the cavity fields and their corresponding driving fields. As a result, the coupled system behaves as a single opto-mechanical system and thus a period-doubling transition to chaos can be realized as one increases the coupling strength. In the case of weak coupling, the dynamics of the two opto-mechanical subsystems are no longer synchronized and the coupled system dynamics unfold in an eight-dimensional phase space. The limit cycles experience the Hopf bifurcation, which leads to the appearance of a toric attractor. Under certain range of parameters, i.e., appropriate frequency detunings, the two-dimensional torus becomes unstable with the increment of coupling strength, and the quasiperiodic transition into chaos occurs in our coupled opto-mechanical system.